User blog:Vel!/Ordinal Hyper-E 2.0
A while ago I successfully formalized BEAF and made it infinitely extensible using some ordinal magic. Not long after, I gave hyper-E notation a shot but I oversimplified the rules and made some mistakes. xE# and E^ are just not as simple as BEAF. E^ stops at epsilon-zero. Aarex tried to extend it to \(\varepsilon_\omega\) with Nested Cascading-E Notation, and inspired by this I revisited E^ and did it right this time. (Someday I'll get a website and this stuff can go in the mainspace.) Update: This is wrong. I will try again later, completely reconsidering my approach. Entries Define \(E_\gamma(\alpha)\): \= \max\{n \in \mathbb{N}_0|\exists \beta_2 < \omega^\gamma, \beta_1: \omega^{\gamma+1} \beta_1 + \omega^\gamma \times n + \beta_2 = \alpha\}\ Also, define \(Q(\alpha) = \{\gamma|E_\gamma(\alpha) \neq 0\}\), which is the set of all positions with non-zero associated entries. Define \(L(\alpha) = \max(Q(\alpha))\), and \(P(\alpha) = \max\{\gamma < L(\alpha)|\gamma \in Q(\alpha)\}\). In other words, \(L(\alpha)\) is the last non-zero entry and \(P(\alpha)\) is the next-to-last one. Prime blocks E^ uses prime blocks, and the prime is the final entry. Let \(q = \min\{q|\alphaq > P(\alpha)\}\). \= \{\alpha[q, \alpha+ 1, \ldots, \alpha+ p - 2\}\] The Three Rules Let \(p = E_{L(\alpha)}(\alpha)\), and let \(\delta = P(\alpha) - L(\alpha)\). Okay, okay, \(\delta = \min\{\beta|L(\alpha) + \beta = P(\alpha)\}\). Sheesh. 1. Base case: If \(\alpha < \omega\), \(S_b(\alpha) = b^{\alpha + 1}\). 2. Default case: If \(\delta = 1\) (i.e. \(P(\alpha) = L(\alpha) + 1\)): \= \sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & \omega^\gamma \times (p - 1) \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\ \= S_b\left(\sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & 0 \\ \gamma = P(\alpha) : & \omega^\gamma \times S_b(\alpha') \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\right)\ :This removes the last entry and replaces the penultimate one with the copy of the array except with the blah blah blah. 3. Hyper-band expansion: If \(\delta\) is of the form \(\omega^\eta \times \omega\): \= S_b\left(\sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & 0 \\ \gamma \in \Pi_p(L(\alpha)) : & \omega^\gamma \times E_{P(\alpha)}(\alpha) \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\right)\ 4. Hyper-product expansion: Otherwise: \= S_b\left(\sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & 0 \\ \gamma = P(\alpha) + \delta[p : & \omega^\gamma \times E_{P(\alpha)}(\alpha) \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\right)\] As you can see, E^ is messier than BEAF, but at least now it works. Example First's let's check a few smaller numbers. Let's try godgahlah = E100#^#100 = \(S_b(\omega^{\omega^\omega}99 + 99)\). \(L(\alpha) = \omega^\omega\), \(P(\alpha) = 0\), and \(p = 99\). So \(\delta = \omega^\omega\) and \(\deltap = 99\). \+ 99) = S_b(\omega^{\omega^{99}}99 + 99)\ which is E100###...(100 times)...###100. Consider tethrathoth = E100#^^#100 = \(S_b(\varepsilon_099 + 99)\). \(L(\alpha) = \varepsilon_0\), \(P(\alpha) = 0\), and \(p = 99\). So \(\delta\) = \(\varepsilon_0\) and \(\deltap = \omega \uparrow\uparrow 99\). \+ 99) = S_b(\omega^{\omega \uparrow\uparrow 99}99 + 99)\ and this is E100#^#^...(100 times)...^#^#100 as expected. Epsilon-one Sbiis's notation got epsilon'd at #^^##. Here's how to fix it. The fundamental sequence for #^^## = \(\varepsilon_1\) is \(\varepsilon_0\), \(\varepsilon_0 \uparrow\uparrow 2\), \(\varepsilon_0 \uparrow\uparrow 3\), ... This suggests the following sequence: #^^# → {#^#^#} (this is Bowers' notation -- the number of #s is not 3, but the prime) (#^^#)^(#^^#) → {#^#^#}^{#^#^#} (#^^#)^(#^^#)^(#^^#) → {#^#^#}^{#^#^#}^{#^#^#} ... #^^## → And for \(\varepsilon_2\): #^^## (#^^##)^(#^^##) (#^^##)^(#^^##)^(#^^##) ... #^^### If we let H be any hyperion expression, the rule is #^^(H*#) → (#^^H)^(#^^H)^...^(#^^H)^(#^^H) p times This rule discreetly defines \(\varepsilon_\alpha\) for successor ordinals \(\alpha\). We can go further with \(\varepsilon_\omega\): #^^(#^#) → #^^#p And the three rules: #^^# → #^#^...^#^# #^^(H*#) → (#^^H)^(#^^H)^...^(#^^H)^(#^^H) p times #^^H → #^^Hp if H is a limit hyperion (Trying to avoid ordinals at this point gets kinda silly. We're staring directly at the fast-growing hierarchy masqueraded in a sea of hash marks!) So now we've defined #^^H for all H. But there's more! #^^(#^^#) is \(\varepsilon_{\varepsilon_0}\), #^^(#^^(#^^#)) is \(\varepsilon_{\varepsilon_{\varepsilon_0}}\), ... Okay, let's cut to the chase and give gamma-zero a shot. #^^^# = {#,#,3} → {#^^#^^#} #^^^^# = {#,#,4} → {#^^^#^^^#} ... <#,#,#> → {#{^^^}#{^^^}#} (limit of Aarex's notation) Our puny bracket notation is leaking, and we're not even at \(\Gamma_0\) yet. Now we have to use BEAF. It seems like everything has to use BEAF! <#,#,<#,#,#>> <#,#,<#,#,<#,#,#>>> <#,#,<#,#,<#,#,<#,#,#>>>> ... <#,#,1,2> Now arriving at Feferman-Schütte, folks! And that's cascading-E notation, taken to expandal arrays. And we can always go further. I will no longer define these notations precisely; the ordinals do just fine. <#,#,#,#> <#,#,#,#,#> <#,#,#,#,#,#> ... <#,#(1)2> I believe this marks the small Veblen ordinal. I'll skip all the multidimensional arrays and go straight to the large Veblen ordinal: <#,#/2> Not to far beyond LVO, the traditional BEAF peters out and we have no more simple ways to notate the ordinals. Further exploration Now that we have a firm foundation, defining #^^^#, #^^^^#, # #, etc. is easy. Just for kicks, I will introduce some mandatory number names: :penthathoth = E100#^^^#100 = \(S(\zeta_099 + 99)\) :hexthathoth = E100#^^^^#100 = \(S(\eta_099 + 99)\) :ulthragahlah = E100# #100 = \(S(\Gamma_099 + 99)\) :horifagahlah = \(S(\vartheta(\Omega^\omega)99 + 99)\) (I still don't know what theta means) :supreme almighty horifagahlah = \(S(\vartheta(\Omega^\Omega)99 + 99)\) :baglaferuncus rex = \(S(\psi(\varepsilon_{\Omega + 1})99 + 99)\) :turagulah = \(S(\omega_1^\text{CK}99 + 99)\) (yeahhh, uncomputable E^) :super turagulah = \(S(\omega_2^\text{CK}99 + 99)\) :ultra turagulah = \(S(\text{fixedPoint}(\alpha \mapsto \omega_\alpha^\text{CK})99 + 99)\) Category:Blog posts